Radiative Corrections to the Direct Detection of Inelastic Scattering of Higgsino-like Neutralino Dark Matter

Technical Summary: Radiative Corrections to the Direct Detection of Inelastic Scattering of Higgsino-like Neutralino Dark Matter

Problem Statement
The Minimal Supersymmetric Standard Model (MSSM) with conserved R-parity offers the lightest neutralino ($\tilde{\chi}^0_1$) as a viable dark matter (DM) candidate. A nearly pure Higgsino-like $\tilde{\chi}^0_1$ is particularly favored for resolving the naturalness problem and accommodating the observed DM relic abundance ($\Omega_{DM}h^2 \simeq 0.12$) via efficient pair-annihilation. However, direct detection (DD) of such Higgsino-like DM is challenging because tree-level couplings to the Higgs and $Z$ bosons are heavily suppressed, leading to negligible spin-independent (SI) scattering cross-sections.

While Next-to-Leading Order (NLO) electroweak (EW) corrections have been computed for elastic scattering ($\tilde{\chi}^0_1 q \to \tilde{\chi}^0_1 q$), a critical aspect often overlooked is the inelastic scattering process ($\tilde{\chi}^0_1 q \to \tilde{\chi}^0_2 q$). In scenarios with a compressed spectrum, the mass splitting ($\delta = m_{\tilde{\chi}^0_2} - m_{\tilde{\chi}^0_1}$) between the lightest neutralino (LSP) and the next-to-lightest neutralino (NLSP) can be extremely small ($\sim \mathcal{O}(10\text{--}100)$ keV). For Galactic DM with typical kinetic energies of $\sim \mathcal{O}(100)$ keV, transitions to the nearly degenerate heavier state become kinematically accessible. Recent observations, such as a nuclear-recoil event near 248 keV by the LUX-ZEPLIN (LZ) experiment, have motivated re-evaluations of TeV-scale Higgsino DM where inelastic channels may dominate or significantly contribute to the total event rate.

Methodology
The authors perform a comprehensive calculation of the SI direct detection cross-sections, incorporating both elastic and inelastic channels with full one-loop EW radiative corrections. The methodology involves:

1. Renormalization Scheme: The authors adopt the on-shell renormalization scheme for the chargino-neutralino sector. This is crucial because the inelastic scattering rate is highly sensitive to the precise value of the mass splitting $\delta$. The scheme ensures that the physical masses of the neutralinos and charginos are correctly defined at the one-loop level.
2. Vertex Corrections: The study computes one-loop corrections to the three-point vertices $\tilde{\chi}^0_1 \tilde{\chi}^0_i h_j$ (where $i, j \in \{1, 2\}$ and $h_j$ are the light and heavy Higgs bosons). This includes:
   • Loop diagrams involving fermions ($F$), scalars ($S$), and vector bosons ($V$), specifically focusing on quark-squark loops (dominated by top-stop loops), chargino-$W$ loops, and neutralino-$Z$ loops.
   • Counterterm contributions derived from wave-function renormalization and mass renormalization.
3. Computational Framework:
   • SPheno and SARAH are used to generate the MSSM spectrum and calculate two-loop Higgs masses and one-loop masses in the $\overline{DR}$ scheme.
   • FeynArts and FormCalc are employed to generate amplitudes for the vertices and reduce them to Passarino-Veltman functions.
   • LoopTools is used for numerical evaluation.
   • The corrected vertices and masses are implemented into micrOMEGAs (v6.3.0) to compute the elastic scattering cross-sections.
   • The inelastic scattering amplitudes are computed using custom routines based on the effective Lagrangian approach, considering only Higgs-mediated scalar operators (neglecting twist-2 and other subdominant contributions for the inelastic channel).
4. Event Rate Calculation: The differential event rates are calculated for the LZ experiment (exposure $4.2$ tonne-yrs), summing contributions from both elastic and inelastic channels. The analysis scans the $M_1$-$M_2$ plane for various Higgsino mass parameters ($\mu = \pm 0.5, \pm 1$ TeV) to identify regions where $\delta \lesssim 100$ keV.

Key Contributions

• Inclusion of Inelastic Channels: Unlike previous studies that focused primarily on elastic scattering or neglected the $\tilde{\chi}^0_1 \tilde{\chi}^0_2 h_j$ vertex corrections, this work explicitly calculates the radiatively corrected inelastic scattering cross-sections.
• On-Shell Mass Splitting: The authors consistently apply the on-shell renormalization scheme to calculate the one-loop corrected masses of the neutralinos and charginos, ensuring a precise determination of the mass splitting $\delta$, which is the governing parameter for inelastic kinematics.
• Interplay of Corrections: The paper demonstrates that NLO corrections can drastically alter the relative magnitude of elastic and inelastic cross-sections. While tree-level couplings for Higgsino-like DM are suppressed, loop-induced corrections can enhance them by orders of magnitude. In specific scenarios, the inelastic cross-section can exceed the elastic cross-section, though the final event rate depends on kinematic suppression.
• Benchmark Analysis: Five benchmark points (BPs) are selected to illustrate different regimes of mass splitting and radiative corrections, comparing Leading Order (LO) and NLO predictions against LZ constraints.

Results

• Magnitude of Corrections: The NLO corrections to the vertex factors ($\tilde{\chi}^0_1 \tilde{\chi}^0_i h_j$) can be substantial, ranging from $\sim 10\%$ to over $500\%$ for certain benchmark points, primarily due to the smallness of the LO values.
• Cross-Section vs. Event Rate: In specific regions of the parameter space (e.g., Benchmark Point 2), the inelastic scattering cross-section at NLO can exceed the elastic cross-section. However, the event rate is subject to Boltzmann suppression due to the minimum velocity required to excite the heavier state ($v_{min}^{inel}$). For BP2, although the inelastic NLO cross-section is comparable to or slightly larger than the elastic NLO cross-section, the inelastic contribution to the total event rate remains slightly lower than the elastic contribution due to this kinematic suppression. For other points (e.g., BP1), the inelastic rate is strongly suppressed relative to the elastic one.
• Boltzmann Suppression: Despite the potential for large cross-sections, the inelastic event rate is often reduced relative to the elastic one because the inelastic channel requires a higher minimum DM velocity to overcome the mass splitting $\delta$.
• LZ Constraints: All benchmark points considered are consistent with the current null results of the LZ experiment at 90% Confidence Level. However, the inclusion of inelastic channels and NLO corrections modifies the predicted event rates, altering the exclusion limits and the required exposure for future detection.
• Loop Contributions: The dominant contributions to the radiative corrections come from top-squark loops ($\tilde{t}q\tilde{q}$ type), followed by chargino-$W$ loops. The heavy Higgs ($h_2$) contributions are suppressed due to its large mass ($> 6$ TeV).

Significance and Claims
The paper claims that a precise theoretical treatment of Higgsino-like DM direct detection is incomplete without considering both elastic and inelastic scattering channels, particularly when the mass splitting is small. The authors emphasize that:

1. Radiative corrections are non-negligible: They can significantly enhance scattering rates, potentially bringing Higgsino-like DM within the reach of current and future experiments like LZ, XENONnT, and PandaX.
2. Inelastic cross-sections can exceed elastic ones: In specific compressed spectrum scenarios, the inelastic cross-section component may exceed the elastic component. However, the total event rate may still be dominated by the elastic channel due to Boltzmann suppression, highlighting the necessity of calculating both the cross-sections and the kinematic factors accurately.
3. Relevance to Recent Data: The framework is directly relevant for interpreting recent LZ nuclear-recoil events (e.g., the 248 keV event) and assessing the viability of TeV-scale Higgsino DM interpretations.
4. Natural SUSY Probes: These studies provide a complementary avenue to probe "natural" supersymmetric scenarios characterized by compressed spectra and suppressed tree-level interactions, which are difficult to access via other experimental means.

The authors conclude that future direct detection experiments, combined with precision theoretical predictions including radiative corrections, are essential for robustly assessing the discovery prospects of Higgsino-like DM.